CHAP, vii] PKIMITIVE ROOTS, EXPONENTS, INDICES. 201 This formula is simplified in the case M = ^(TW) and the numbers belonging to this Haupt-exponent are called primitive roots of m. The primitive roots of m divide into families of <£OK?w)) each, such that any two of one family are powers of each other modulo m, while no two of different families are powers of each other. Each family is subdivided. In general, not every integer prime to m occurs among the residues modulo m of the powers of the various primitive roots of m. A. Cunningham113 considered the exponent £ to which an odd number q belongs modulo 2m; and gave the values of £ when m^ 3, and when q = 2*12=*= 1 (0 odd), m>3. When q=2x=rl and m>x+l, the residue of gl/2y can usually be expressed in one of the forms 1 =p2tt, 1 =F2tt=p2^. G. Fontene*114 determined the numbers N which belong to a given exponent pm~hd modulo pm, where 6 is a given divisor of p~l, and A^l, without employing a primitive root of pm. If p>2, the conditions are that N shall belong to the exponent 5 modulo p and that the highest power of p dividing N' — l shall be ph, l^h^m. *M. Demeczky115 discussed primitive roots. E. Landau116 proved the existence of primitive roots of powers of odd primes, discussed systems of indices for any modulus n, and treated the characters of n. G. A. Miller117 noted that the determination of primitive roots of g corresponds to the problem of finding operators of highest order in the cyclic group G of order g. By use of the group of isomorphisms of G it is shown that the primitive roots of g which belong to an exponent 2q, where q is an odd prime, are given by —a2, when a ranges over those integers between 1 and g/2 which are prime to g. As a corollary, the primitive roots of a prime 2g-f-l, where q is an odd prime, are —a2, l<a<g+l. A. N. Korkine118 gave a table showing for each prime p< 4000 a primitive root g and certain characters which serve to solve any solvable congruence afea (mod p), where q is a prime dividing p —1. Let qa be the highest power of q dividing p—1. The characters of degree q are the solutions of u'sl, u«=u, u"q^uf,. .., (w««-10<rstt(-a) (mod p) and hence are the residues of the powers of </p~1)/Q for k = 1,. .., a. There are noted some errors in the Canon of Jacobi23 and the table of Burckhardt. Korkine stated that if p is a prime and a belongs to the exponent e = (p—1)/6, exactly <£(p~l)/4>(e) of the roots of rc*=a (mod p) are primitive roots of p. K. A. Posse119 remarked that Korkine constructed his table without knowing of the table by Wertheim,91 and extended Korkine's tables to 10000. "'Messenger of Math., 37, 1907-8, 162-4. »«Nouv. Ann. Math., (4), 8, 1908, 193-216. 118Math. 6s Phys. Lapok, Budapest, 17, 1908, 79-86. 118Handbuch.. .Verteilung der Primzahlen, I, 1909, 391-414, 478-486. 117Amer. Jour. Math., 31, 1909, 42-4. "8Matem. Shorn. Moskva (Math. Soc. Moscow), 27, 1909, 28-115, 120-137 (in Russian). Cf. D. A. Grave, 29, 1913, 7-11. The table was reprinted by Posse.1" llfl/m, 116-120, 175-9, 238-257. Reprinted by Posse.1"